On the characterization of expansion maps for self-affine tilings

We consider self-affine tilings in ${\mathbb{R}}^n$ with expansion matrix $\varphi$ and address the question which matrices $\varphi$ can arise this way. In one dimension, $\lambda$ is an expansion factor of a self-affine tiling if and only if $\mathrm{\mid }\lambda \mathrm{\mid }$ is a Perron number, by a result of Lind. In two dimensions, when $\varphi$ is a similarity, we can speak of a complex expansion factor, and there is an analogous necessary condition, due to Thurston: if a complex $\lambda$ is an expansion factor of a self-similar tiling, then it is a complex Perron number. We establish a necessary condition for $\varphi$ to be an expansion matrix for any $n$, assuming only that $\varphi$ is diagonalizable over the complex numbers. We conjecture that this condition on $\varphi$ is also sufficient for the existence of a self-affine tiling.